Galois group: 16T13

• Label: 16T13
• Degree: $16$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_{8}$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$13$
Group:		$D_{8}$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$16$
Generators:		(1,11)(2,12)(3,10)(4,9)(5,8)(6,7)(13,15)(14,16), (1,4,6,8,10,12,14,15)(2,3,5,7,9,11,13,16)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $D_{8}$ x 2

Low degree siblings

8T6 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 2)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
	$ 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$
	$ 8, 8 $	$2$	$8$	$( 1, 4, 6, 8,10,12,14,15)( 2, 3, 5, 7, 9,11,13,16)$
	$ 4, 4, 4, 4 $	$2$	$4$	$( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,16)( 4, 8,12,15)$
	$ 8, 8 $	$2$	$8$	$( 1, 8,14, 4,10,15, 6,12)( 2, 7,13, 3, 9,16, 5,11)$
	$ 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$

Group invariants

Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$
Label:		16.7
Character table:

		1A	2A	2B	2C	4A	8A1	8A3
	Size	1	1	4	4	2	2	2
	2 P	1A	1A	1A	1A	2A	4A	4A
	Type
16.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.7.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
16.7.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
16.7.1d	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.7.2a	R	$2$	$2$	$0$	$0$	$-2$	$0$	$0$
16.7.2b1	R	$2$	$-2$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
16.7.2b2	R	$2$	$-2$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$